Energy efficiency in multiaccess fading channels under QoS constraints

Transcription

1 Univrsity of braska - Lincoln DigitalCommons@Univrsity of braska - Lincoln Faculty Publications from th Dpartmnt of Elctrical and Computr Enginring Elctrical & Computr Enginring, Dpartmnt of Enrgy fficincy in multiaccss fading channls undr QoS constraints Dli Qiao Univrsity of braska-lincoln, dlqiao@c.cnu.du.cn Mustafa Cnk Gursoy Univrsity of braska-lincoln, gursoy@ngr.unl.du Snm Vlipasalar Univrsity of braska-lincoln, svlipasalar@unl.du Follow this and additional works at: Part of th Computr Enginring Commons, and th Elctrical and Computr Enginring Commons Qiao, Dli; Cnk Gursoy, Mustafa; and Vlipasalar, Snm, "Enrgy fficincy in multiaccss fading channls undr QoS constraints". Faculty Publications from th Dpartmnt of Elctrical and Computr Enginring This Articl is brought to you for fr and opn accss by th Elctrical & Computr Enginring, Dpartmnt of at DigitalCommons@Univrsity of braska - Lincoln. It has bn accptd for inclusion in Faculty Publications from th Dpartmnt of Elctrical and Computr Enginring by an authorizd administrator of DigitalCommons@Univrsity of braska - Lincoln.

2 RESEARCH Opn Accss Enrgy fficincy in multiaccss fading channls undr QoS constraints Dli Qiao *, Mustafa Cnk Gursoy and Snm Vlipasalar Abstract In this articl, transmission ovr multiaccss fading channls undr quality-of-srvic QoS constraints is studid in th low-powr and widband rgims. QoS constraints ar imposd as limitations on th buffr violation probability. Th ffctiv capacity, which charactrizs th maximum constant arrival rats in th prsnc of such statistical QoS constraints, is mployd as th prformanc mtric. A two-usr multiaccss channl modl is considrd, and th minimum bit nrgy lvls and widband slop rgions ar charactrizd for diffrnt transmission and rcption stratgis, namly tim-division multipl-accss TDMA, suprposition coding with fixd dcoding ordr, and suprposition coding with variabl dcoding ordr. It is shown that th minimum rcivd bit nrgis achivd by ths diffrnt stratgis ar th sam and indpndnt of th QoS constraints in th lowpowr rgim, whil thy vary with th QoS constraints in th widband rgim. Whn widband slop rgions ar considrd, th suboptimality of TDMA with rspct to suprposition coding is provn in th low-powr rgim. On th othr hand, it is shown that TDMA in th widband rgim can intrstingly outprform suprposition coding with fixd dcoding ordr. Th impact of varying th dcoding ordr at th rcivr undr crtain assumptions is also invstigatd. Ovrall, nrgy fficincy of diffrnt transmission stratgis undr QoS constraints ar analyzd and quantifid. Kywords: ffctiv capacity, nrgy fficincy, nrgy pr information bit, low-powr rgim, multipl-accss fading channls, quality of srvic, suprposition coding, tim-division multipl accss, widband rgim, widband slop. Introduction Enrgy fficincy is an important considration in wirlss systms and has bn rigorously analyzd from an information-thortic prspctiv. In [], Vrdú has xtnsivly studid th spctral fficincy-bit nrgy tradoff in th widband rgim, considring th Shannon capacity as th prformanc mtric. For th Gaussian multiaccss channl, Cair t al. [] hav shown that tim division multipl-accss TDMA is in gnral a suboptimal transmission schm in th low-powr rgim unlss on considrs th asymptotic scnario in which th powr vanishs. It is also shown that fading channl maks th suprposition stratgy mor favorabl. In this analysis, Shannon capacity formulation is again adoptd as th main prformanc mtric. Howvr, Shannon * Corrspondnc: dqiao76@huskrs.unl.du Dpartmnt of Elctrical Enginring, Univrsity of braska-lincoln, E 68588, USA Full list of author information is availabl at th nd of th articl capacity dos not quantify th prformanc in th prsnc of quality-of-srvic QoS limitations in th form of constraints on quuing dlays or quu lngths. Indd, most communication- and information-thortic studis, whil providing powrful rsults, do not gnrally concntrat on dlay and QoS constraints [3]. At th sam tim, providing QoS guarants is on of th ky rquirmnts in th dvlopmnt of nxt gnration wirlss communication ntworks sinc data traffic with multimdia contnt is xpctd to grow significantly and in ral-tim multimdia applications, such as voic ovr IP VoIP and intractiv-vido.g., vidoconfrncing, latncy is a ky QoS mtric. Many fforts hav bn mad to incorporat th dlay constraints in th prformanc analysis [4-7]. In [4], dlay limitd capacity has bn proposd as a prformanc mtric, which is dfind as th rat that can b attaind rgardlss of th valus of th fading stats. In [6], th tradoff btwn th avrag transmission powr and avrag dlay has bn analyzd Qiao t al; licns Springr. This is an Opn Accss articl distributd undr th trms of th Crativ Commons Attribution Licns which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd.

3 Pag of 6 by considring an optimization problm in which th wightd combination of th avrag powr and avrag dlay is minimizd ovr transmission policis that dtrmin th transmission rat by taking into account th arrival stat, buffr occupancy, channl stat jointly togthr. In [7], th long-trm avrag capacity has bn proposd to study th fading multiaccss channl in th widband rgim and th suboptimality of TDMA has bn shown again. In this articl, w follow a diffrnt approach. W considr statistical QoS constraints and study th nrgy fficincy undr such limitations. W adopt th notion of ffctiv capacity [8], which can b sn as th maximum constant arrival rat that a givn tim-varying srvic procss can support whil providing statistical QoS guarants. Th analysis and application of ffctiv capacity in various sttings hav attractd much intrst rcntly s.g. [9-6] and rfrncs thrin. For instanc, rlatd to this study, in [3,4], nrgy fficincy is addrssd in a singl-usr stting whn th wirlss systms oprat undr buffr constraints and mploy ithr adaptiv or fixd transmission schms for point-to-point links. Th ffctiv capacity rgions for multiaccss channl with diffrnt schduling policis hav bn charactrizd in [6]. In that work, it has bn found that TDMA and suprposition coding with variabl dcoding with rspct to channl stats can outprform suprposition stratgy with fixd dcoding. In this articl, w considr th prformanc of TDMA and suprposition stratgy in th prsnc of statistical QoS constraints but concntrat on th low-sr rgim. Mor spcifically, w mploy th tools providd in [,] to invstigat th bit nrgy and widband slop rgions undr QoS constraints in th low-powr and widband rgims. Th main contributions of this articl ar summarizd in th following: W show that diffrnt transmission and rcption stratgis do not affct th minimum bit nrgy lvls rquird by ach usr. Additionally, w prov that whil th minimum bit nrgis ar indpndnt of th QoS constraints in th low powr rgim, thy vary with th QoS constraints in th widband rgim. W dtrmin that suprposition coding with variabl dcoding ordr dos not improv th prformanc in trms of slop rgion with rspct to fixd dcoding ordr in th low powr rgim, whil it can achiv largr slop rgion in th widband rgim. 3 Whn widband slop rgions ar considrd, w show that TDMA is always suboptimal in th low powr rgim xcpt th spcial cas in which fading stats ar linarly dpndnt. On th othr hand, TDMA in crtain cass is dmonstratd to prform bttr than suprposition coding with fixd dcoding ordr in th widband rgim. W also idntify th condition for TDMA to b suboptimal in this rgim. Th rmaindr of th articl is organizd as follows. In Sction, th systm modl is brifly discussd. Sction 3 prsnts som prliminaris on th analysis tools and ffctiv capacity. Th rsults in th low-powr rgim ar providd in Sction 4. Sction 5 prsnts th rsults in th widband rgim. Finally, Sction 6 concluds this articl.. Systm modl As shown in Figur, w considr an uplink scnario whr M usrs with individual powr and buffr constraints i.., QoS constraints communicat with a singl rcivr. It is assumd that th transmittrs gnrat data squncs which ar dividd into frams of duration T. Ths data frams ar initially stord in th buffrs bfor thy ar transmittd ovr th wirlss channl. Th discrt-tim signal at th rcivr in th ith symbol duration is givn by Y [i] = M h j [i] X j [i] n [i], i =,,... j= whr M is th numbr of usrs, X j [i] and h j [i] dnot th complx-valud channl input and th fading cofficint of th jth usr, rspctivly. W assum that {h j [i]} s ar jointly stationary and rgodic discrt-tim procsss, and w dnot th magnitud-squar of th fading cofficints by z j [i] = h j [i].ltz =z,z,..., z m b th channl stat vctor. Abov, n[i] is a zro-man, circularly symmtric, complx Gaussian random variabl with varianc E { n[i] } =. Th additiv Gaussian nois sampls {n[i]} ar assumd to form an indpndnt and idntically distributd i.i.d. squnc. Finally, Y [i] dnots th rcivd signal. Th channl input of usr j is subjct to an avrag nrgy constraint E { x j [i] } P j /B for all j, whrb is th bandwidth availabl in th systm. Assuming that th symbol rat is B complx symbols pr scond, w can s that this formulation indicats that usr j is subjct to an avrag powr constraint of P j.withths Figur Th systm modl.

4 Pag 3 of 6 dfinitions, th avrag transmittd signal to nois ratio of usr j is SRj = P j B. 3. Prliminaris 3.. Effctiv capacity rgion of th MAC channl In [8], ffctiv capacity is dfind as th maximum constant arrival rat that a givn srvic procss can support in ordr to guarant a statistical QoS rquirmnt spcifid by th QoS xponnt θ. Th ffctiv capacity is formulatd as C θ = lim t θt log E { θs[t]} bits/s, whr th xpctation is with rspct to S [t] = t i= s [i], which is th tim-accumulatd srvic procss, and {s[i], i =,,...} dnots th discrt-tim stochastic srvic procss. Effctiv capacity can b rgardd as th maximum throughput of th systm whil th buffr violation probability is guarantd to dcay xponntial fast with dcay rat controlld by θ, i.., th buffr violation probability bhavs as Pr {Q > Q max } θq max for larg Qmax,whr Q is stationary quu lngth. W assum that th fading cofficints stay constant ovr th fram duration T and vary indpndntly from on fram to anothr for ach usr. Hnc, w basically considr a block-fading modl. In this scnario, s[i] = TR [i], whr R[i] is th instantanous srvic rat in th ith fram duration [it; it. Thn, th ffctiv capacity in can b xprssd as C θ = θt log E { z θtr[i] } bits/s, 3 whr R[i] is in gnral a function of th fading stat z. 3 is obtaind using th fact thatinstantanousrats{r [i]} vary indpndntly from on fram to anothr. It is intrsting to not that as θ and hnc QoS constraints rlax, ffctiv capacity approachs th rgodic capacity, i.., C θ E z {R [i]}. On th othr hand, as shown in [3], C θ convrgs to th dlay limitd capacity as θ grows without limit i.., θ andqosconstraints bcom incrasingly mor strict. Thrfor, ffctiv capacity nabls us to study th prformanc lvls btwn th two xtrm cass of dlay limitd capacity, which can b sn as a dtrministic srvic guarant or quivalntly as a prformanc lvl attaind undr hard QoS limitations, and rgodic capacity, which is achivd in th absnc of any QoS considrations. Suppos that Θ =θ,...,θ M is a vctor composd of th QoS constraints of M usrs. Lt β j = θ jtb log, j =,,..., M b th associatd normalizd QoS constraints. Also, lt CΘ =C θ,..., C M θ M dnot th vctor of th normalizd ffctiv capacitis. In [6], th ffctiv capacity rgions of th multiaccss channl for diffrnt schduling policis hav bn charactrizd. Th ffctiv capacity rgion achivd by TDMA is C : C j θj θ {δj} j TB log E δjθjtblog SR j δ j zj whr δ j is th fraction of tim allocatd to usr j. Th ffctiv capacity rgion achivd by suprposition coding with fixd dcoding ordr is givn by {τm} { { C : C j θj θ j TB log E z θjt }} M! m= τmr πm j whr τ m is th fraction of tim allocatd to a spcific dcoding ordr π m, R π m j rprsnts th maximal instantanous srvic rat of usr j at a givn dcoding ordr π m, which is givn by R π m j = B log SR j z j πm i>πm j SR 6 iz i whr πm is th invrs trac function of π m. Dcoding ordrs can b varid for ach channl fading stat z. Suppos th vctor spac R M of th possibl valus for z is partitiond into M! disjoint rgions {Z m } M! m= with rspct to dcoding ordrs {π m} M! m=. Thn, th maximum ffctiv capacity that can b achivd by th jth usr is C j θj = θ j TB log E { } z θ j TR j = θ j TB log M! m= z Z m θ j TR π m jp z z dz for j =,..., M,whrp z is th distribution function of z and R π m j is givn in Spctral fficincy vs. bit nrgy If w dnot th ffctiv capacity normalizd by bandwidth or quivalntly th spctral fficincy in bits pr scond pr Hrtz by C E SR, θ = C E SR, θ B = θtb log E { θtr[i]}, 8 thn it can b asily sn that E b undr QoS constraints can b obtaind from []

5 Pag 4 of 6 E b = lim SR SR C E SR = Ċ E. 9 Hnc, nrgy fficincy improvs as SR diminishs and th minimum bit nrgy is attaind as SR vanishs. At E b,thslops of th spctral fficincy vrsus E b / indbcurvisiscalldthwidband slop, and is dfind as [] S = lim E b E b C E Eb E log b E log b log. Considring th xprssion for normalizd ffctiv capacity, th widband slop can b found from a S = Ċ E log C E whr Ċ E and C E ar th first and scond drivativs, rspctivly, of th function C E SR in bits/ s/hz at zro SR []. Th minimum bit nrgy E b and th widband slop provid a linar approximation of th spctral fficincy-bit nrgy curv at low SR lvls and nabls us to charactriz and quantify th nrgy fficincy in th low-sr rgim. 4. Enrgy fficincy in th low-powr rgim As dscribd abov, in ordr to transmit nrgy fficintly and achiv bit nrgy lvls clos to th minimum lvl, on nds to oprat in th low-sr rgim in which ithr th powr is low or bandwidth is larg. In this sction, w considr th low-powr rgim. W concntrat on th two-usr multiaccss channl. Blow, w first not th maximum ffctiv capacitis attaind through diffrnt transmission stratgis dscribd in in Sction. Subsquntly, w idntify th corrsponding minimum bit nrgis and th widband slops. ow, for th two-usr TDMA, if w fix th fraction of tim allocatd to usr as δ Î [, ], th maximum ffctiv capacitis of th two-usrs in th TDMA rgion givn by 4 bcom C SR = θ TB log E z and C SR = θ TB log E δθtb log z rspctivly, { δθ TB log SR z δ SR z δ }, 3 xt, considr suprposition coding with fixd dcoding ordr. W dnot th ratio of th transmittr-sid signal-to-nois ratios as λ = SR SR = P B / P.W B assum that th valu of this ratio is arbitrary but is kpt fixd as SR and SR diminish in th low-sr rgim. Additionally, w lt τ dnot th fraction of tim in which th dcoding ordr, is mployd. ot that if th dcoding ordr is,, th rcivr first dcods th scond usr s signal in th prsnc of intrfrnc from first usr s signal, and subsquntly dcods th first usr s signal with no intrfrnc. ot that th symmtric cas occurs whn th dcoding ordr is, in th rmaining -τ fraction of th tim. Whn this stratgy is usd, th maximum ffctiv capacitis in th rgion dscribd in 5 can now b xprssd as C SR SR = z θtb τ log θ log TB Ez SRzτ log SR z /λ C SR SR = z θtb τ log θ TB log E z τ log SRz λsr z, 4.5 Finally, w turn our attntion to suprposition coding with variabl dcoding ordr. In this cas, th dcoding ordr dpnds on th fading cofficints z, z. W dfin z = gsr =glsr as th partition function in th z - z spac. b Dpnding on which dcoding ordr is mployd in ach rgion, w hav diffrnt ffctiv capacity xprssions. If usrs ar dcodd in th ordr, whn z <gsr and ar dcodd in th ordr, whn z >gsr, th ffctiv capacitis ar givn by C SR = θ TB log gsr θtb log SRz p z z, z dz dz gsr θtb log SRz SRz/λ 6 p z z, z dz dz, C SR = θ TB gλsr log θtb log SRz p z z, z dz dz SR z θtb log λsr z p z z, z dz dz. gλsr 7

6 Pag 5 of 6 Similar ffctiv capacity xprssions can b drivd if usrs ar dcodd in th ordr, if z <gsr and dcodd in th ordr, if z >gsr. Assumption : Throughout th articl, w considr th partition functions gsr thatsatisfythfollowing proprtis: g is finit. Th first and scond drivativs of g with rspct to SR, ġ SR and g SR, xist. Morovr, ġ and g ar finit. As will b sn in th nsuing analysis, th finitnss assumptions abov will srv as sufficint conditions to nsur that th drivativs of ffctiv capacity in th limit as SR vanishs ar finit. Dnot E b,i = SR i as th bit nrgy of usr i =,. C i Th rcivd bit nrgy is E r b,i = E b,i E {z i }. 8 As th following rsult shows, th minimum rcivd bit nrgis for th diffrnt stratgis ar th sam. Thorm : For all λ = SR SR and all gz, SR satisfying th proprtis in Assumption, th minimum rcivd bit nrgy for th multiaccss fading channl attaind through TDMA, suprposition coding with fixd dcoding ordr, or suprposition dcoding with varying dcoding ordr, is th sam and is givn by E r b, = Er b, =log =.59 db. 9 Proof: S Appndix. Rmark : Th rsult of Thorm shows that diffrnt transmission stratgis.g., TDMA or suprposition coding and diffrnt rcption schms.g., fixd or variabl dcoding ordrs lad to th sam fundamntal limit on th minimum bit nrgy. Similarly as in [], TDMA is optimally fficint in th asymptotic rgim in which th signal-to-nois ratio vanishs. Mor intrstingly, w not that this rsult is obtaind in th prsnc of QoS constraints. Additionally, th minimum bit nrgy is clarly indpndnt of th QoS limitations paramtrizd by th QoS xponnts θ and θ. Hnc, th nrgy fficincy is not advrsly affctd by th buffr constraints in this asymptotic rgim in which SR. Rmark : It can b asily shown using th ffctiv capacity xprssions providd in 4, 5, and 7 that th charactrization in Thorm, i.., th rsult that th minimum rcivd nrgy pr bit rquirmnt for ach usr is -.59 db undr QoS constraints, holds in amorgnralsttinginwhichthnumbrofusrs M. Having shown that th minimum bit nrgis achivd by diffrnt transmission and rcption stratgis ar th sam for ach usr, w not that th widband slop rgions hav bcom mor intrsting sinc thy quantify th prformanc in th non-asymptotic rgim in which SRs ar small but nonzro. With th analysis approach introducd in [], w hav th following rsults. Thorm : Th multiaccss slop rgion achivd by TDMA is givn by S = { S, S : S S up, S S up κ κ κ } κ κ κ κ S κ S whr S up = S up = E {z } { } β E z E {z } E { } z, E {z } { } β E z E {z } E { z }, E {z } κ = { } β E z E {z }, E { z } κ = { } β E z E {z }, E {z } κ = { } β E z E {z }, E { } z κ = { } β E z E {z },, b = θ TBlog and b = θ TBlog. Proof: S Appndix. Th following rsults provid th widband slop xprssions whn suprposition transmission is mployd. Thorm 3: For any λ = SR SR, th multiaccss slop rgion achivd by th suprposition coding with fixd dcoding ordr is S = { S, S : S S up, S S up, λe {z } E {z z } S S up E {z } λe {z z } S S up } =, whr S up and S up ar th sam as dfind in Thorm. Proof: S Appndix 3. Thorm 4: For any λ = SR SR, and any gsr satisfying th proprtis in Assumption, th multiaccss

7 Pag 6 of 6 slop rgion achivd by suprposition coding with variabl dcoding ordr is S = { S, S : S S up, S S up, λe {z } E {z z } S S up E {z } λe {z z } S S up } =, whr S up and S up ar th sam as dfind in Thorm. Proof: S Appndix 4. Rmark 3: Comparing 63 with 65 or 64 with 66 in th proof of Thorm 4 in Appndix 4, w s that diffrnt dcoding ordrs do not chang th widband slop valus for givn usr only if g = z, i.., th z -z spac is qually dividd. On mor intrsting rmark is that if w compar th third conditions in and, w notic that fixd dcoding ordr achivs th sam prformanc as variabl dcoding ordr. Rmark 4: It is intrsting to not in th abov rsults that, unlik th minimum bit nrgy lvls, th widband slops dpnd on th QoS xponnts θ and θ through b and b. Indd, as can b sn from th xprssions of th uppr bounds S up and S up, th widband slops tnd to diminish as QoS constraints bcom mor stringnt and θ and θ incras. Smallr slops indicat that at a givn nrgy pr bit lvl gratr than Er b, a smallr spctral fficincy is attaind. Thrfor, spctral fficincy dgrads undr mor strict QoS constraints. Equivalntly, to achiv th sam lvl of spctral fficincy, highr nrgy pr bit is rquird. Hnc, from this prspctiv, a pnalty in nrgy fficincy is xprincd as buffr limitations bcom mor stringnt. In th following rsult, w stablish th suboptimality of TDMA. Thorm 5: Th widband slop rgion of TDMA is insid th on attaind with suprposition coding. Proof: W only nd to considr th third conditions of and. Substituting 58 and 59 into th lfthandsidlhsofththirdconstraintin,w obtain E { } z κ κ E { z } τ E{z z } λ E { } z E { z } λτe{z z }. 3 Comparing th sum of th last two trms with or mor prcisly subtracting from th sum, w can writ E { } z E { } z E { z } τ E { z E{z z } } λτe{z z } λ E { { } z = } E z 4τE{z z } 4E{z z } τ 4 E { } z τ E { } E{z z } z λτe{z z }. λ W ar intrstd in th numrator which is a quadratic function of th paramtr τ. W not that th discriminant of this quadratic function satisfis =6E {z z } 4 6E {z z } E { z } { } E z =6E {z z } E {z z } E { z } { } 5 E z whr th Cauchy-Schwarz inquality E {z z } E { z } E { z } is usd. Thus, th numrator of 4 is always nonngativ, i.., th slop rgion achivd by TDMA is insid th on achivd by suprposition coding. Th quality holds only if z and z ar linarly dpndnt. In Figur, w plot th slop rgions in indpndnt Rayligh fading channls with variancs E {z } = E {z } =. W assum b =andb =. From th figur, w immdiatly obsrv th suboptimality of TDMA compard with suprposition coding. 5. Enrgy fficincy in th widband rgim In this sction, w considr th widband rgim in which th ovrall bandwidth of th systm B is larg. Lt ζ =. Similar as in [3], w know that th minimum bit B nrgy achivd in spars multipath fading channls c as B or quivalntly ζ can b xprssd as E b,i P i ζ / = lim ζ C i ζ = P i/, i =,. 6 Ċ i To mak th analysis mor clar, blow w first xprss th capacity xprssions in -7 as functions of ζ. and 3 can b rwrittn as C ζ = ζ θ T log E z Slop of Usr TDMA δθ T ζ =.5 log P z ζ δ = Slop of Usr, 7 Suprposition Figur Th slop rgions for indpndnt Rayligh fading channls.

8 Pag 7 of 6 and C ζ = ζ θ T log E z δ θ T ζ log P z ζ δ, 8 rspctivly. For suprposition coding with fixd dcoding ordr, and fixd λ = SR = P ζ / = P, 4 and 5 now SR P ζ / P bcom C ζ P θ z ζ T P z ζ τ log τlog ζ = ζ P z ζ θ T log E z C ζ P θ z ζ T τ log ζ = ζ P z ζ θ T log E z P z ζ τlog., 9 3 ot that w can writ gsr as g P ζ, so similarly w can writ 6 and 7 as functions of ζ C ζ = ζ θt Pzζ θ log ζ log p T z z, z dz dz Pζ g P Pζ θt z ζ g ζ log Pzζ p z z, z dz dz Pζ g C ζ = ζ θt Pzζ θ log ζ log p z z, z dz dz T θt ζ log Pζ g P z ζ Pzζ p z z, z dz dz. 3 3 Thn w immdiatly hav th following rsult. Thorm 6: For all gsr satisfying th proprtis in Assumption, th minimum bit nrgis for th twousr multiaccss fading channl in th widband rgim attaind through TDMA, suprposition coding with fixd dcoding ordr, and suprposition dcoding with varying dcoding ordr, dpnd on th individual QoS constraints at th usrs and ar givn by E b, = E b, = θ TP θ, TP log E z log z θ TP θ, TP log E z log z rspctivly. Proof: S Appndix 5. Rmark 5: As Thorm 6 shows, th sam minimum bit nrgy is achivd through diffrnt transmission stratgis.g., TDMA or suprposition coding and diffrnt rcption schms.g., fixd or variabl dcoding ordrs, and thrfor TDMA is optimally nrgy fficint in th widband rgim as B. As bfor, Thorm 6 can b radily xtndd and similar xprssions for th minimum nrgy pr bit can b asily obtaind for cass in which thr ar mor than usrs, i.., M. Rmark 6: A stark diffrnc from th rsult in Thorm is that th minimum bit nrgy now varis with th spcific QoS constraints at th usrs. Whn θ =, w can immdiatly show that th right-hand sids of 33 and 34 bcom log E{z } and log E{z, rspctivly, which } is quivalnt to 9. For θ >, th nrgy fficincy is now advrsly affctd by th buffr constraints in th widband rgim. Similarly as in Sction 4, w nxt invstigat th widband slops in ordr to quantify th prformancs and nrgy fficincis of diffrnt transmission and rcption mthods in th non-asymptotic rgim in which th bandwidth B is larg but finit. W hav th following rsults. Thorm 7: In th widband rgim, th multiaccss slop rgion achivd by TDMA is givn by { S = S, S : S S up, S S up, S S up S } S up 35

9 Pag 8 of 6 whr S up log = θ TP S up log = θ TP θ TP E z log z log E z θ TP E z E z log z z θ TP E z log z log E z θ TP θ TP log z, log z z θ TP log z. Proof: S Appndix 6. Thorm 8: In th widband rgim, th multiaccss slop rgion achivd by suprposition coding with fixd dcoding ordr is S = { S, S : S S up log, S Sup, θ T θt P log Ez log z θt P E z log z θt P P P E z log z z z log S S up θ T θt P log Ez log z θt P E z log z θt P S S up = P P E z log z z z 36 whr S up and S up ar dfind in Thorm 7. Proof: S Appndix 7. Thorm 9: For any gsr satisfying th proprtis in Assumption, th multiaccss slop rgions achivd by suprposition coding with variabl dcoding ordr in th widband rgim ar diffrnt for diffrnt dcoding ordrs. Th slop rgion is S = {S, S : {g} log S θ T θtp log Ez log z θtp E z log z P θtp Ez log z z θtp g P P log z z z p z, z dz dz log S θ T θtp log Ez log z θtp E z log z P θtp θ TP Ez log z z P P g log z z z p z, z dz dz 37 if th dcoding ordr is, whn z <gz,sr, and th dcoding ordr is, whn z >gz, SR. Th slop rgion is S = {S, S : {g} log S θ T θtp log Ez log z θtp E z log z P θtp θ TP Ez log z z P P g log z z z p z, z dz dz log S θ T θtp log Ez log z θtp E z log z P θtp Ez log z z θtp g P P log z z z p z, z dz dz 38 if th dcoding ordr is, whn z <gz,sr, and th dcoding ordr is, whn z >gz, SR. Proof: S Appndix 8. Rmark 7: Unlik prvious discussions, w hav no closd form xprssion for th widband slop rgion achivd by suprposition coding with variabl dcoding ordr in th widband rgim. Anothr obsrvation in th abov rsult is that diffrnt dcoding ordrs can rsult in diffrnt widband slop rgions. Blow w show th supriority of suprposition coding with variabl dcoding compard with fixd dcoding ordr. Thorm : Suprposition coding with variabl dcoding ordr achivs bttr prformanc in trms of widband slop rgion with rspct to suprposition coding with fixd dcoding ordr. Proof: S Appndix 9. Inthfollowing,wprsntthconditionundr which th suboptimality of TDMA compard with suprposition coding with fixd dcoding ordr can b stablishd. Thorm : If th following is satisfid E z θ T P log z θ T P E z z z log z E z z θ T P log z E z z z θ 39 T P log z z, thn th widband slop rgion of TDMA is insid th on attaind with suprposition coding with fixd dcoding ordr. Proof: W considr th third conditions in 35 and 36. Substituting 86 and 87 into th LHS of th third condition in 35, w hav

10 Pag 9 of 6 θ TP τ P E z log z z z θ TP θ TP P E z log z z τ E z log z z z P E z θ TP P E z log z z log z z τp E z θ TP θ TP log z z z 4 So if th widband slop rgion is insid th on attaind with suprposition coding with fixd dcoding ordr, w must hav th abov valu to b gratr than for all τ. Aftr subtracting from 4, w can obtain P θ TP P E z log z z τ E z 4E z P θ TP P E z log z z τp E z θ TP log z z z E z θ TP 4E z θ TP log z z z θ TP log z z z θ TP log z z z log z z z θ TP E z log z z E z τ θ TP E z log z z z τ θ TP log z z 4 th widband slop rgion attaind with suprposition coding with fixd dcoding ordr. This tlls us that TDMA can b a bttr choic compard with suprposition coding with fixd dcoding ordr in som cass. As an additional point, w not that if, on th othr hand, th condition in 39 is satisfid, TDMA prforms wors than suprposition coding with variabl dcoding ordr as wll du to th charactrization in Thorm. In th numrical rsults, w plot th widband slop rgions for indpndnt Rayligh fading channls with variancs E {z } = E {z } =.Wassumθ =.,θ =., T =ms.infigur3,wassum P = P = 4. Th LHS of 39 is.9, whil th right-hand sid is.83. Hnc, th inquality is satisfid. From th figur, w can s that TDMA is suboptimal compard with suprposition coding. In Figur 4, w assum P = P = 4. Th LHS of 39 is.3, whil th right-hand sid is.6. Hnc, th inquality is not satisfid. Confirming th abov discussion, w can obsrv in th figur that TDMA indd achivs points outsid th slop rgion attaind with suprposition coding with fixd dcoding ordr. 6. Conclusion In this articl, w hav analyzd th nrgy fficincy of two-usr multiaccss fading channls undr QoS constraints by mploying th ffctiv capacity as a masur of th maximal throughput. W hav charactrizd th minimum bit nrgy and th widband slop rgions for diffrnt transmission stratgis. W hav conductd our analysis in two rgims: low-powr rgim and widband rgim. Through this analysis, w hav shown th impact of QoS constraints on th nrgy fficincy of multiaccss Th first two trms of th multiplication ar positiv valus. Th minimum valu of th third trm which is a quadratic function of τ is achivd at τ =,andth minimum valu is E z θ T P log z θ T P E z z E z log z z z θ T P log z θ T P E z z 4 log z z z Slop of usr.5.5 TDMA Suprposition with fixd dcoding ordr Thus, w obtain th condition statd in 39 for TDMA to b suboptimal. Rmark 8: It is intrsting that if th condition 39 is not satisfid, TDMA can achiv som points outsid Slop of usr Figur 3 Th slop rgions for indpndnt Rayligh fading channls.

11 Pag of 6 Substituting 43 and 44 into 9, w hav Slop of usr TDMA Suprposition with fixd dcoding ordr E b, = log min E {z }, 45 E b, = log 46 min E {z } which imply 9 according to 8. For th suprposition coding with fixd dcoding, valuating th first drivativ of 4 and 5 at SR = and SR =, w immdiatly obtain Slop of usr Figur 4 Th slop rgions for indpndnt Rayligh fading channls. Ċ = E {z } log Ċ = E{z } log fading channls. Mor spcifically, w hav found that th minimum bit nrgis ar th sam for ach usr whn diffrnt transmission and rcption tchniqus ar mployd. Whil ths minimum valus ar qual thos that can b attaind in th absnc of QoS constraints in th low-powr rgim, w hav shown that strictly highr bit nrgy valus, which dpnd on th QoS constraints, ar ndd in th widband rgim. W hav also sn that whil TDMA is suboptimal in th low-powr rgim whn widband slop rgions ar considrd, it can outprform suprposition coding with fixd dcoding ordr in th widband rgim. Morovr, w hav provn in th widband rgim that varying th dcoding ordr can achiv largr slop rgion whn compard with fixd dcoding ordr for suprposition coding. umrical rsults validating our rsults ar providd as wll. Appndix. Proof of Thorm Considr th TDMA stratgy. Taking th first drivativ of th functions in and 3 and ltting SR =, SR =, w obtain Ċ = E {z } log, 43 Ċ = E {z } log. 44 which again imply 9 taking into considration 9 and 8. xt,wprovthrsultforthvariabldcoding cas. First, w considr 6 and 7 with th associatd dcoding ordr assignmnt. Th first drivativ of 6 can b xprssd as φ Ċ SR = β φ log = SR z β β φ log β gsr β gsr p z, g SR ġ SR dz SR z β z p z, z dz z SR z β SR g SR /λ p z, g SR ġ SR dz SR z β SR z /λ z SR z /λ p z, z dz dz 49 whr φ is th first drivativ of j, which is dfind as

12 Pag of 6 φ = gz,sr θtb log SRz p z z, z dz dz gz,sr θtb log SR z SR z /λ p z z, z dz dz. 5 Undr th assumptions that g and ġ ar finit, w can asily s from 49 that ltting SR = lads to Ċ = E {z } log. 5 Similarly, taking th first drivativ of 7 and ltting SR =, w obtain Ċ = E {z } log. 5 Applying th dfinitions 9 and 8, w prov 9 for this dcoding ordr assignmnt. For th rvrs dcoding ordr assignmnt i.., usrs ar dcodd in th ordr, if z <gsr anddcoddinthordr, if z >gsr, following similar stps, w again obtain th rsult in 9.. Proof of Thorm Taking th scond drivativs of th functions in and 3 and ltting SR =, SR =, w obtain C = β E {z } E { z } log δ E { z } 53 and C = β E {z } E { z } log δ E { z }. 54 Combining 43, 44, 53, and 54 with, w now gt E {z } S = { } β E z E {z } δ E { z } 55 E {z } S = { } β E z E {z } δ E { z } 56 which, aftr liminating δ, provid us th third condition in. 3. Proof of Thorm 3 Th scond drivativs of th functions 4 and 5 at zro signal-to-nois ratio ar C = β E {z } β E { z log C = } τ λ βe {z } β E { z } λτe {z z }. log E {z z } 57 Thn, th widband slops ar givn by E {z } S = { } β E z E {z } E { z } τ E {z z } 58 λ E {z } S = { } β E z E {z } E { } z λτe {z z }. 59 Aftr solving for τ in 58 and 59 and subtracting th rsulting quations, w obtain th third condition in 4. Proof of Thorm 4 W nd to considr th widband slops for diffrnt dcoding ordr assignmnts. Du to th complx xprssions involvd, w hr stat th drivation for S for th cas in which th dcoding ordr is, whn z <gz, SR, and th dcoding ordr is, whn z >gz, SR. Taking th scond drivativ of 6, w hav C SR = φ φ φ β φ log 6 whr φ is providd in 49 and φ is givn by φ = β SR z β p z, g SR g SR dz SR g SR /λ SR z SR g SR /λ β z SR g SR /λ p z, g SR ġ SR dz SR z βṗ z, g SR ġ SR dz SR g SR /λ β β β λ β gsr SR z SR z /λ β z SR g SR /λ 4 p z, z dz dz SR z SR z /λ β z z SR g SR /λ 3 p z, z dz dz SR z β p z, g SR g SR dz SR z β z p z, g SR ġ SR dz SR z β ṗ z, g SR ġ SR dz β β gsr SR z β z p z, z dz dz. 6 Ltting SR = and supposing that g, ġ,and g ar finit, w hav

13 Pag of 6 C = { } β E z log E {z} E { z } g z z p z, z dz dz λ Substituting 6 and 5 into, w obtain E {z } S = { } β E z E {z } E { z } λ Similarly, w can driv E {z } S = { } β E z E {z } E { z } λ 6 g z z p z, z dz dz. 63 g zzp z, z dzdz. 64 If th dcoding ordr is, whn z <gz,sr, and is, whn z >gz,sr, following th stps dscribd abov, w can obtain E {z } S = { } β E z E {z } E { z } λ E {z } S = { } β E z E {z } E { z } λ g zzp z, z dzdz 65 g z z p z, z dz dz. 66 Combining 63 and 64 and liminating g, w can obtain th third condition in. It is intrsting that combining 65 and 66 and liminating g, w still gt th sam third condition statd in. This shows us that th slop rgions for diffrnt dcoding ordr assignmnts ovrlap. 5. Proof of Thorm 6 Taking th first drivativs of 7 and 8 and ltting ζ =, w obtain Ċ = θ T log E z Ċ = θ T log E z θ TP log z θ TP log z Substituting 67 and 68 into 6, w gt th rsults in 33 and 34. xt, w considr th suprposition coding with fixd dcoding. Evaluating th first drivativ of 9 and 3 at ζ =, w again gt Ċ = θ T log E z θ TP log z 69 Ċ = θ T log E z θ TP log z. 7 which imply th rsults in 33 and 34. W can also prov th rsults for th variabl dcoding cas similarly as in th proof of Thorm. Considr 3 and 3 with th associatd dcoding ordr. Th first drivativ of 3 can b xprssd as Ċ ζ = θ T log φ ζ φ θ Tφ 7 whr j is φ = P ζ g θ T P ζ g and φ is Pζ φ = ġ ζ log θt P θ T P z ζ ζ log Pzζ ζ log θt Pzζ ζ log Pζ g θ T ζ log Pzζ θt ζ ġ P ζ θt P ζ log Pζ θt g ζ log θt ζ p z z, z dz dz P z ζ P z ζ p z z, z dz dz p z z, g P ζ / dz P z log Pzζ P z ζ Pg P ζ / ζ P z ζ Pzζ P z log Pzζ Pzζ θ T ζ log Pzζ pz z, z dzdz p z z, g P ζ / dz P z ζ Pzζ p z z, z dz dz. If w dfin f ζ = θ T ζ log P z ζ θ T ζ w can show that 7 73 P z log P z ζ

14 Pag 3 of 6 log Pzζ Pz log ζ Pzζ lim f ζ = θt lim ζ ζ ζ Pz Pz = θt lim ζ ζ log Pzζ log ζ Pzζ Pzζ log = lim f ζ θt Pz ζ log which givs us that lim f ζ = ζ θ T log P z assignmnt, following similar stps, w still gt th rsults in 33 and Proof of Thorm 7 Th scond drivativs of 7 and 8 at ζ = ar E z C = P δlog E z θ TP log z z θ TP log z 8 Similarly, w can show that P z ζ θ T lim ζ ζ log P θt z ζ ζ θ T P z = θt P P z z log log. P z ζ P z log P z ζ P z ζ With 75 and 76 in mind, w can obtain 76 C = δ log P θ TP E z log z z θ 8 TP E z log z lim φ = ζ θ T log and hnc θ T log E z g θ T P log z P z θ T P log z P P z z p z, z dz dz 77 Using th dfinition in, w can xprss th widband slops as E z log S =δ θ TP θtp log z log Ez θtp E z log z z θtp log z 8 Ċ = θ T log E z θ TP log z. 78 Similarly, taking th drivativ of 3 and ltting ζ =, w hav θtp E z log z θtp log S =δ log Ez log z θ TP θtp E z log z z 83 Ċ = θ T log E z θ TP log z. 79 which, aftr incorporating 6, again givs us th rsults in 33 and 34. For th rvrs dcoding ordr which aftr simpl computation giv us th third condition in Proof of Thorm 8 Evaluating th scond drivativs of 9 and 3 at ζ = yilds

15 Pag 4 of 6 θtp P log z z τ θtp PP Ez C = Ez log z zz log θtp log z Ez P θtp E z log z z τpp θtp C = E z log z z z log θtp E z log z and as a rsult, th widband slops ar givn by log S = θt θtp log Ez log z θtp log z Ez P θtp Ez log z θtp z τ PPEz log z zz θt Pzζ Pζ P ζ φ = g log p z, g Pζ / dz θt Pzζ Pζ P ζ log ġ θt ζ log Pzζ θt Pz log p z, g Pζ / dz ζ Pzζ θt Pzζ Pζ P ζ ġ log θt ζ log Pzζ θt Pz log ṗ z, g Pζ / dz ζ Pzζ θt Pzζ ζ log θt ζ log Pzζ θt Pz log Pζ ζ Pzζ g θt ζ ζ log Pzζ θt Pz log ζ Pzζ θt Pz p z, z ζ log Pzζ dzdz Pzζ θt ζ log Pζ Pg Pζ / ζ P g p z, g Pζ / dz Pzζ θt ζ log Pζ Pg Pζ / ζ P ġ θt Pzζ ζ log PgPζ /ζ 89 θtp log Ez log z θtp E z log z log S = θ T P θtp Ez log z θtp z τp P E z log z z z 87 Aftr solving for τ in 86 and 87 and subtracting th rsulting quations, w hav th third condition in Proof of Thorm 9 Similar to Thorm 4, w hr prsnt th drivation for S for th cas whn th dcoding ordr is, whn z <gz, SR, and th dcoding ordr is, whn z >g z, SR. Th scond drivativ of 3 is C ζ = φ θ Tφ ζ whr j and φ is givn by φ φ φ θ Tφ 88 φ ar 7 and 73, rspctivly, and θt Pz log p z, g Pζ / dz ζ PgPζ /ζ Pzζ PgPζ /ζ Pzζ θt ζ log Pζ Pg Pζ / ζ P ġ ṗ z, g Pζ / dz Pzζ Pζ θt g ζ log Pzζ θt Pzζ ζ log Pzζ θt Pz log ζ Pzζ Pzζ Pzζ θt Pzζ ζ ζ log θt Pz log Pzζ ζ Pzζ Pzζ Pzζ θt Pz Pz log Pzζ Pzζ Pz Pz Pzζ ζ Pzζ Pzζ Pzζ p z, z dzdz By ltting ζ =andrcalling75and76,wcan show that C = log P Ez θtp log z z PP θtp g log z zzp z, z dzdz θtp log z Ez 9

16 Pag 5 of 6 Combining 78 and 9 with, w hav log S = θ T log Ez θt P log z θt P P Ez log z z P P θt P E z log z θt. P g log z z z p z, z dz dz 9 θtp g log z z z p z, z dz dz θtp E z log z z z θtp log Ez log z log = θ T θtp P P E z E z log z z z θtp log z S S up 95 Following similar stps, w can driv that and log S = θ T log Ez θt P P Ez log z z P P θt P log z θt P E z log z. θ T P g log z z z p z, z dz dz 9 If th dcoding ordr is, whn z <gz,sr, and is, whn z >gz,sr, following th stps dscribd abov, w can obtain log S = θ T log Ez θt P P Ez log z z P P log S = θ T log Ez θt P log z θt P E z log z, θ T P g log z z z p z, z dz dz θt P log z θt P P Ez log z z P P θt P E z log z θt. P g log z z z p z, z dz dz Also not that th widband slops hav non-ngativ valus and w hav th inqualitis in 37 and Proof of Thorm W nd to compar th uppr bound of th slop rgion in 36 with th uppr bounds of both 37 and 38. By moving th trm with g to th LHS of th quation, w can rwrit 9 and 9 as θ TP g log z z z p z, z dz dz θtp E z log z z z θtp log Ez log z log = θ T θtp P P E z Dnot γ = γ = E z log z z z θtp log z S S up θ TP g log z z z p z, z dz dz θ TP E z log z z z g E z θ TP log z z z p z, z dz dz θ TP log z z z W know that g and g varywith diffrnt g. Substitut 95 and 96 into th third condition of 36, w can obtain θtp log Ez log z θtp E z log z log θ T θtp S S up P P E z log z z z θtp log Ez log z θtp E z log z log θ T θtp S S up P P E z log z z z = γ γ 99

17 Pag 6 of 6 Following similar stps, w can gt from 93 and 94 θtp log Ez log z θtp E z log z log θ T θtp S S up P P E z log z z z θtp log Ez log z θtp E z log z log θ T θtp S S up P P E z log z z z = γ γ Considring 99 and, w know that ithr g g or - g - g must b lss than, which implis that variabl dcoding ordr achivs points outsid th rgion attaind with fixd dcoding ordr, proving th thorm. Endnots a W not that th xprssions in 9 and diffr from thos in [] by a constant factor du to th fact that w assum that th units of C E is bits/s/hz rathr than nats/s/hz. b Th partition function can in gnral b a function of z as wll, i.., gsr =gz,sr. c As discussd in [3,4], widband and low-powr rgims ar quivalnt if rich multipath fading is xprincd. Hnc, in such a cas, th sam minimum bit nrgy and widband slop xprssions ar obtaind in both rgims. 6. RA Brry, RG Gallagr, Communication ovr fading channls with dlay constraints. IEEE Trans Inf Thory. 48, doi:.9/ D Tunintti, G Cair, S Vrdú, Fading multi-accss channls in th widband rgim: th impact of dlay constraints, in Proc IEEE Int Symp Information Thory, Lausann, Switzrland, 94 Jun/July 8. D Wu, R gi, Effctiv capacity: a wirlss link modl for support of quality of srvic. IEEE Trans Wirl Commun. 4, J Tang, X Zhang, Quality-of-srvic drivn powr and rat adaptation ovr wirlss links. IEEE Trans Wirl Commun. 68, J Tang, X Zhang, Cross-layr-modl basd adaptiv rsourc allocation for statistical QoS guarants in mobil wirlss ntworks. IEEE Trans Wirl Commun. 76, L Liu, P Parag, J Tang, W-Y Chn, J-F Chambrland, Rsourc allocation and quality of srvic valuation for wirlss communication systms using fluid modls. IEEE Trans Inf Thory. 535, L Liu, P Parag, J-F Chambrland, Quality of srvic analysis for wirlss usrcoopration ntworks. IEEE Trans Inf Thory. 53, MC Gursoy, D Qiao, S Vlipasalar, Analysis of nrgy fficincy in fading channl undr QoS constrains. IEEE Trans Wirl Commun. 88, D Qiao, MC Gursoy, S Vlipasalar, Th impact of QoS constraints on th nrgy fficincy of fixd-rat wirlss transmissions. IEEE Trans Wirl Commun. 8, D Qiao, MC Gursoy, S Vlipasalar, Enrgy fficincy in th low-sr rgim undr quuing constraints and channl uncrtainty. IEEE Trans Commun. 597, D Qiao, MC Gursoy, S Vlipasalar, Transmission stratgis in multipl accss fading channls with statistical QoS constraints. IEEE Trans Inf Thory. 583, doi:.86/ Cit this articl as: Qiao t al.: Enrgy fficincy in multiaccss fading channls undr QoS constraints. EURASIP Journal on Wirlss Communications and tworking :36. Acknowldgmnts This study was supportd by th ational Scinc Foundation undr Grants CS and CCF Author dtails Dpartmnt of Elctrical Enginring, Univrsity of braska-lincoln, E 68588, USA Dpartmnt of Elctrical Enginring and Computr Scinc, Syracus Univrsity, Syracus, Y 344, USA Compting intrsts Th authors dclar that thy hav no compting intrsts. Rcivd: Octobr Accptd: 6 April Publishd: 6 April Rfrncs. S Vrdú, Spctral fficincy in th widband rgim. IEEE Trans Inf Thory. 486, doi:.9/tit G Cair, D Tunintti, S Vrdú, Suboptimality of TDMA in th low-powr rgim. IEEE Trans Inf Thory. 54, doi:.9/ TIT A Ephrmids, B Hajk, Information thory and communication ntworks: an unconsummatd union. IEEE Trans Inf Thory. 44, doi:.9/ SV Hanly, DC Ts, Multiaccss fading channls-part II: dlay-limitd capacitis. IEEE Trans Inf Thory. 447, doi:.9/ IE Tlatar, RG Gallagr, Combining quuing thory with information thory for multiaccss. IEEE J Sl Aras Commun. 3, doi:.9/ Submit your manuscript to a journal and bnfit from: 7 Convnint onlin submission 7 Rigorous pr rviw 7 Immdiat publication on accptanc 7 Opn accss: articls frly availabl onlin 7 High visibility within th fild 7 Rtaining th copyright to your articl Submit your nxt manuscript at 7 springropn.com